Antilinear deformations of Coxeter groups, an …fring/Micro.pdf · Introduction Invariant Calogero...

Introduction Invariant Calogero models Antilinear deformation of Coxeter groups Deformed Calogero models Conclusions

Antilinear deformations of Coxeter groups,an application to Calogero models

Andreas Fring

Indian Statistical InstituteKolkata 21/04/2010

based on: A.F. and Monique Smith, arXiv:1004.0916

Antilinear deformations of Coxeter groups,an application to Calogero models

Andreas Fring

Indian Statistical InstituteKolkata 21/04/2010

based on: A.F. and Monique Smith, arXiv:1004.0916

Outline

1 Introductiongeneral motivationPT-symmetry as an example of an antilinear operator

2 Possibilities to deform Calogero modelsaddition of PT-symmetric termrestriction of KdV sollutionsdeformation of dynamical variables⇔ def. of root space

3 Antilinear deformations of Coxeter groups4 An application to Calogero models5 Conclusions

General motivation

Operators O which are left invariant under an antilinearinvolution I and whose eigenfunctions Φ also respect thissymmetry,

[O, I] = 0 ∧ IΦ = Φ

have a real eigenvalue spectrum.[E. Wigner, J. Math. Phys. 1 (1960) 409]

By defining a new metric also a consistent quantummechanical framework has been developed for theoriesinvolving such operators.In particular this also holds for O being non-Hermitian.

General motivation

Operators O which are left invariant under an antilinearinvolution I and whose eigenfunctions Φ also respect thissymmetry,

[O, I] = 0 ∧ IΦ = Φ

have a real eigenvalue spectrum.[E. Wigner, J. Math. Phys. 1 (1960) 409]

By defining a new metric also a consistent quantummechanical framework has been developed for theoriesinvolving such operators.In particular this also holds for O being non-Hermitian.

PT-symmetry as an example of an antilinear involution

Unbroken PT -symmetry guarantees real eigenvalues (QM)

PT -symmetry: PT : x → −x p → p i → −i(P : x → −x ,p → −p; T : x → x ,p → −p, i → −i)

PT is an anti-linear operator:

PT (λΦ + µΨ) = λ∗PT Φ + µ∗PT Ψ λ, µ ∈ C

Real eigenvalues from unbroken PT -symmetry:

[H,PT ] = 0 ∧ PT Φ = Φ ⇒ ε = ε∗ for HΦ = εΦ

Proof :εΦ = HΦ = HPT Φ = PT HΦ = PT εΦ = ε∗PT Φ = ε∗Φ

Examples of non-Hermitian models in the literatureLattice Reggeon field theory

[J.L. Cardy, R. Sugar, Phys. Rev. D12 (1975) 2514]Quantum spin chains

[G. von Gehlen, J. Phys. A24 (1991) 5371]strings on AdS5 × S5-background

[A. Das, A. Melikyan, V. Rivelles, JHEP 09 (2007) 104]affine Toda field theory (β ∈ iR)deformed space-time structures

[B. Bagchi and A.F., Phys. Lett. A373 (2009) 4307]dynamical noncommutative space-time

[A.F., L. Gouba and F. Scholtz, arXiv:1003.3025 ]

Adding PT-symmetric terms

Extended Calogero models

HBK =p2

∑iq2

∑i 6=k

1(qi − qk )2 +i g

∑i 6=k

1(qi − qk )

with g, g ∈ R,q,p ∈ R`+1 (similarly for Bn)[B. Basu-Mallick, A. Kundu, Phys. Rev. B62 (2000) 9927]

Generalized Hamiltonian:

Hµ =12

p2 +12

∑α∈∆

g2αV (α · q) + iµ · p

· µ = 1/2∑

α∈∆ gαf (α · q)α, f (x) = 1/x V (x) = f 2(x)· includes also Sutherland-Moser potentials· ∆ is any root system, i.e. algebra independent· representation independent[A. F., Mod. Phys. Lett. A21 (2006) 691, Acta P. 47 (2007) 44]

HBK =p2

∑iq2

∑i 6=k

1(qi − qk )2 +i g

∑i 6=k

1(qi − qk )

Hµ =12

p2 +12

∑α∈∆

g2αV (α · q) + iµ · p

· µ = 1/2∑

HBK =p2

∑iq2

∑i 6=k

1(qi − qk )2 +i g

∑i 6=k

1(qi − qk )

Hµ =12

p2 +12

∑α∈∆

g2αV (α · q) + iµ · p

· µ = 1/2∑

- Not so obvious that one can re-write

Hµ =12

(p+iµ)2+12

∑α∈∆

g2αV (α·q), g2

s + α2s g2

s α ∈ ∆sg2

l + α2l g2

l α ∈ ∆l

⇒ Hµ = η−1hCalη with η = e−q·µ

- integrability follows trivially in L = [L,M]: L(p)→ L(p + iµ)

- computing backwards for any CMS-potential

Hµ =12

p2 +12

∑α∈∆

g2αV (α · q) + iµ · p − 1

- µ2 = α2s g2

s∑α∈∆s

V (α · q) +α2l g2

l∑α∈∆l

V (α · q) only for V rational

Hµ =12

(p+iµ)2+12

∑α∈∆

g2αV (α·q), g2

s + α2s g2

s α ∈ ∆sg2

l + α2l g2

l α ∈ ∆l

Hµ =12

p2 +12

∑α∈∆

g2αV (α · q) + iµ · p − 1

- µ2 = α2s g2

s∑α∈∆s

V (α · q) +α2l g2

l∑α∈∆l

Hµ =12

(p+iµ)2+12

∑α∈∆

g2αV (α·q), g2

s + α2s g2

s α ∈ ∆sg2

l + α2l g2

l α ∈ ∆l

Hµ =12

p2 +12

∑α∈∆

g2αV (α · q) + iµ · p − 1

- µ2 = α2s g2

s∑α∈∆s

V (α · q) +α2l g2

l∑α∈∆l

Hµ =12

(p+iµ)2+12

∑α∈∆

g2αV (α·q), g2

s + α2s g2

s α ∈ ∆sg2

l + α2l g2

l α ∈ ∆l

Hµ =12

p2 +12

∑α∈∆

g2αV (α · q) + iµ · p − 1

- µ2 = α2s g2

s∑α∈∆s

V (α · q) +α2l g2

l∑α∈∆l

Deformed KdV-systems/Calogero models (Particle-field duality)

From real fields to complex particle systems

i) No restrictionse.g. Benjamin-Ono equation

ut + uux + λHuxx = 0 (*)

H ≡ Hilbert transform, i.e. Hu(x) = Pπ

∫∞−∞

u(x)z−x dz

u(x , t) =λ

∑k=1

x − zk− i

x − z∗k

)∈ R

satisfies (*) iff zk obeys the An-Calogero equ. of motion

zk =λ2

∑k 6=j

(zj − zk )−3

[H. Chen, N. Pereira, Phys. Fluids 22 (1979) 187][talk by J. Feinberg, PHHQP workshop VI, 2007, London ]

ii) restrict to submanifoldTheorem: [Airault, McKean, Moser, CPAM, (1977) 95 ]Given a Hamiltonian H(x1, . . . , xn, x1, . . . , xn) with flow

xi = ∂H/∂xi and xi = −∂H/∂xi i = 1, . . . ,n

and conserved charges Ij in involution with H,i.e.Ij ,H = 0. Then the locus of grad I = 0 is invariant.Example: Boussinesq equation

vtt = a(v2)xx + bvxxxx + vxx (**)

Thenv(x , t) = c

k=1(x − zk )−2

satisfies (**) iff b=1/12, c=-a/2 and zk obeys

zk = 2∑

j 6=k(zj − zk )−3 ⇔ zk = −∂H

zk = 1−∑

j 6=k(zj − zk )−2 ⇔ grad(I3 − I1) = 0

ii) restrict to submanifoldTheorem: [Airault, McKean, Moser, CPAM, (1977) 95 ]Given a Hamiltonian H(x1, . . . , xn, x1, . . . , xn) with flow

xi = ∂H/∂xi and xi = −∂H/∂xi i = 1, . . . ,n

and conserved charges Ij in involution with H,i.e.Ij ,H = 0. Then the locus of grad I = 0 is invariant.Example: Boussinesq equation

vtt = a(v2)xx + bvxxxx + vxx (**)

Thenv(x , t) = c

k=1(x − zk )−2

satisfies (**) iff b=1/12, c=-a/2 and zk obeys

zk = 2∑

j 6=k(zj − zk )−3 ⇔ zk = −∂H

zk = 1−∑

j 6=k(zj − zk )−2 ⇔ grad(I3 − I1) = 0

[P. Assis and A.F., J. Phys. A42 (2009) 425206]

Calogero-Moser-Sutherland models (deformed)

HCMS =p2

∑α∈∆s

(α ·q)2 +12

∑α∈∆

gαV (α ·q) m,gα ∈ R

- invariance with respect to Coxter groupW

HCMS =σip · σip

∑α∈∆s

(α · σiq)2 +12

∑α∈∆

gαV (α · σiq)

∑α∈∆s

(σ−1i α · q)2 +

∑α∈∆

gαV (σ−1i α · q)

- aim: construct new models which are invariant underWPT

HPT CMS =p2

∑α∈∆s

(α·q)2+12

∑α∈∆

gαV (α·q), m,gα ∈ R

A2,G2: [A. F., M. Znojil, J. Phys. A41 (2008) 194010]all Coxeter groups: [A. F., Monique Smith, arXiv:1004.0916]

Calogero-Moser-Sutherland models (deformed)

HCMS =p2

∑α∈∆s

(α ·q)2 +12

∑α∈∆

gαV (α ·q) m,gα ∈ R

- invariance with respect to Coxter groupW

HCMS =σip · σip

∑α∈∆s

(α · σiq)2 +12

∑α∈∆

gαV (α · σiq)

∑α∈∆s

(σ−1i α · q)2 +

∑α∈∆

gαV (σ−1i α · q)

- aim: construct new models which are invariant underWPT

HPT CMS =p2

∑α∈∆s

(α·q)2+12

∑α∈∆

gαV (α·q), m,gα ∈ R

A2,G2: [A. F., M. Znojil, J. Phys. A41 (2008) 194010]all Coxeter groups: [A. F., Monique Smith, arXiv:1004.0916]

General strategy

Construction of antilinear deformationsInvolution ∈ W ≡ Coxeter group⇒ deform in antilinear wayFind a linear deformation map:

δ : ∆→ ∆(ε) α 7→ α = θεα

αi ∈ ∆ ⊂ Rn, αi(ε) ∈ ∆(ε) ⊂ Rn ⊕ ıRn, ε ∈ R∆(ε) remains invariant under an antilinear transformation ω

(i) ω : α = µ1α1 + µ2α2 7→ µ∗1ωα1 + µ∗2ωα2 for µ1, µ2 ∈ C(ii) ω2 = I(iii) ω : ∆→ ∆.

Candidates:- Weyl reflections: σi ∈ W- factors of Coxeter element: σ± ∈ W- longest element: ω0 ∈ W

General strategy

δ : ∆→ ∆(ε) α 7→ α = θεα

General strategy

δ : ∆→ ∆(ε) α 7→ α = θεα

General strategy

δ : ∆→ ∆(ε) α 7→ α = θεα

PT-symmetrically deformed Coxeter factors

• simple Weyl reflections:

σi(x) := x − 2x · αi

α2iαi , with 1 ≤ i ≤ ` ≡ rankW,

• general Coxeter transformations:

σ =∏`

i=1σi ,

• special Coxeter transformations:

σ := σ−σ+, σ± :=∏

i∈V±

σi , [σi , σj ] = 0 for i , j ∈ V±

A` :+−+−+

· · ·u u u u u• deformed Coxeter element factors:

σε± := θεσ±θ−1ε = τσ± : ∆(ε)→ ∆(ε),

τ ≡ complex conjugation

σi(x) := x − 2x · αi

σ =∏`

i=1σi ,

σ := σ−σ+, σ± :=∏

i∈V±

σi , [σi , σj ] = 0 for i , j ∈ V±

A` :+−+−+

σε± := θεσ±θ−1ε = τσ± : ∆(ε)→ ∆(ε),

σi(x) := x − 2x · αi

σ =∏`

i=1σi ,

σ := σ−σ+, σ± :=∏

i∈V±

σi , [σi , σj ] = 0 for i , j ∈ V±

A` :+−+−+

σε± := θεσ±θ−1ε = τσ± : ∆(ε)→ ∆(ε),

σi(x) := x − 2x · αi

σ =∏`

i=1σi ,

σ := σ−σ+, σ± :=∏

i∈V±

σi , [σi , σj ] = 0 for i , j ∈ V±

A` :+−+−+

σε± := θεσ±θ−1ε = τσ± : ∆(ε)→ ∆(ε),

• deformed Coxeter element:

σε := θεσθ−1ε = σε−σ

ε+ = τ2σ−σ+ = σ : ∆(ε)→ ∆(ε),

⇒[σ, θε] = 0

• deformed Coxeter orbits:

Ωεi :=

γ i , σεγ i , σ

2ε γ i , . . . , σ

h−1ε γ i

= θεΩi

with γ i = ci αi , ci = ± for i ∈ V±• deformed root space:

∆(ε) :=⋃`

i=1Ωε

i = θε∆(ε)

• invariance:

σε± : ∆(ε)→ θεσ±θ−1ε ∆(ε) = θεσ±∆(ε) = θε∆(ε) = ∆(ε)

• limit limε→0:

limε→0

αi(ε) = αi limε→0

∆(ε) = ∆

ε+ = τ2σ−σ+ = σ : ∆(ε)→ ∆(ε),

⇒[σ, θε] = 0

Ωεi :=

2ε γ i , . . . , σ

h−1ε γ i

= θεΩi

∆(ε) :=⋃`

i=1Ωε

i = θε∆(ε)

• invariance:

limε→0

∆(ε) = ∆

ε+ = τ2σ−σ+ = σ : ∆(ε)→ ∆(ε),

⇒[σ, θε] = 0

Ωεi :=

2ε γ i , . . . , σ

h−1ε γ i

= θεΩi

∆(ε) :=⋃`

i=1Ωε

i = θε∆(ε)

• invariance:

limε→0

∆(ε) = ∆

ε+ = τ2σ−σ+ = σ : ∆(ε)→ ∆(ε),

⇒[σ, θε] = 0

Ωεi :=

2ε γ i , . . . , σ

h−1ε γ i

= θεΩi

∆(ε) :=⋃`

i=1Ωε

i = θε∆(ε)

• invariance:

limε→0

∆(ε) = ∆

ε+ = τ2σ−σ+ = σ : ∆(ε)→ ∆(ε),

⇒[σ, θε] = 0

Ωεi :=

2ε γ i , . . . , σ

h−1ε γ i

= θεΩi

∆(ε) :=⋃`

i=1Ωε

i = θε∆(ε)

• invariance:

limε→0

∆(ε) = ∆

• take θε to be an isometry:

αi · αj = αi · αj .

⇒θ∗ε = θ−1

ε and det θε = ±1

Summary: properties of θε(i) θ∗εσ± = σ±θε

(ii) [σ, θε] = 0(iii) θ∗ε = θ−1

(iv) det θε = ±1(v) limε→0 θε = I

Solutions?

⇒θ∗ε = θ−1

(ii) [σ, θε] = 0(iii) θ∗ε = θ−1

Solutions?

⇒θ∗ε = θ−1

(ii) [σ, θε] = 0(iii) θ∗ε = θ−1

Solutions?

∵ [σ, θε] = 0, make Ansatz:

θε =∑h−1

k=0ck (ε)σk , lim

ε→0ck (ε) =

1 k = 00 k 6= 0

, ck (ε) ∈ C

⇒ into θ∗εσ± = σ±θε (c0 = r0, ch/2 = rh/2, ck = ırk otherwise)

θε =

r0(ε)I + ı

(h−1)/2∑k=1

rk (ε)(σk − σ−k ) h odd,

r0(ε)I + rh/2(ε)σh/2 + ıh/2−1∑k=1

rk (ε)(σk − σ−k ) h even.

⇒ into det θε = ±1

±1 =∏

[r0(ε)− 2

(h−1)/2∑k=1

rk (ε) sin(2πk

h sn)]

±1 =∏

[r0(ε) + (−1)nrh/2(ε)− 2

h/2−1∑k=1

rk (ε) sin(2πk

h sn)]

h even

sn are the exponents of a particularW

θε =∑h−1

k=0ck (ε)σk , lim

ε→0ck (ε) =

1 k = 00 k 6= 0

, ck (ε) ∈ C

θε =

r0(ε)I + ı

(h−1)/2∑k=1

r0(ε)I + rh/2(ε)σh/2 + ıh/2−1∑k=1

±1 =∏

[r0(ε)− 2

(h−1)/2∑k=1

rk (ε) sin(2πk

h sn)]

±1 =∏

[r0(ε) + (−1)nrh/2(ε)− 2

h/2−1∑k=1

rk (ε) sin(2πk

h sn)]

h even

θε =∑h−1

k=0ck (ε)σk , lim

ε→0ck (ε) =

1 k = 00 k 6= 0

, ck (ε) ∈ C

θε =

r0(ε)I + ı

(h−1)/2∑k=1

r0(ε)I + rh/2(ε)σh/2 + ıh/2−1∑k=1

±1 =∏

[r0(ε)− 2

(h−1)/2∑k=1

rk (ε) sin(2πk

h sn)]

±1 =∏

[r0(ε) + (−1)nrh/2(ε)− 2

h/2−1∑k=1

rk (ε) sin(2πk

h sn)]

h even

Now case-by-case∆(ε) for A3

θε = r0I + r2σ2 + ır1

(σ − σ3

)with explicit representation

−1 0 01 1 00 0 1

, σ2 =

1 1 00 −1 00 1 1

1 0 00 1 10 0 −1

, σ =

−1 −1 01 1 10 −1 −1

σ− = σ1σ3, σ+ = σ2, σ = σ−σ+

θε =

r0 − ır1 −2ır1 −ır1 − r22ır1 r0 − r2 + 2ır1 2ır1

−ır1 − r2 −2ır1 r0 − ır1

all constraints require

(r0 + r2)[(r0 + r2)2 − 4r2

r0 − r2 + 2r1 = (r0 − r2 + 2r1) (r0 + r2)

(r0 + r2) = (r0 − r2)2 − 4r21

these are solved by

r0(ε) = cosh ε, r1(ε) = ±√

cosh2 ε− cosh ε, r2(ε) = 1−cosh ε

⇒ simple deformed roots

α1 =cosh εα1 + (cosh ε− 1)α3−ı√

cosh ε sinh( ε

)(α1+2α2+α3) ,

α2 =(2 cosh ε− 1)α2 + 2ı√

cosh ε sinh( ε

)(α1 + α2 + α3) ,

α3 =cosh εα3 + (cosh ε− 1)α1−ı√

cosh ε sinh( ε

)(α1+2α2+α3) .

remaining positive rootsα4 := α1 + α2, α5 := α2 + α3, α6 := α1 + α2 + α3.

∆(ε) for A4n−1-subseriesclosed solution

θε = r0I + r2nσ2n + ırn

(σn − σ−n) ,

- with r2n = 1− r0, rn = ±√

r20 − r0

- useful choice r0 = cosh ε∆(ε) for E6

θε =

r0 −2ır2 0 −2ır2 −2ır2 −ır22ır2 r0 + ır2 2ır2 2ır2 2ır2 2ır20 2ır2 r0 + 2ır2 4ır2 3ır2 2ır2−2ır2 −2ır2 −4ır2 r0 − 5ır2 −4ır2 −2ır22ır2 2ır2 3ır2 4ır2 r0 + 2ır2 0−ır2 −2ır2 −2ır2 −2ır2 0 r0

r2 = ±1/

√3√

r20 − 1 , r0 = cosh ε

∆(ε) for B2n+1-subseriesno solution

(σn − σ−n) ,

- with r2n = 1− r0, rn = ±√

r20 − r0

θε =

r2 = ±1/

√3√

r20 − 1 , r0 = cosh ε

(σn − σ−n) ,

- with r2n = 1− r0, rn = ±√

r20 − r0

θε =

r2 = ±1/

√3√

r20 − 1 , r0 = cosh ε

Solutions from folding

for instance: Bn → A2n-deformed A6-roots:

α1 = cosh εα1 − ı/√

7 sinh ε(α1 + 2α2 + 2α3 + 2α4 − 2α6)

α2 = cosh εα2 + ı/√

7 sinh ε(2α1 + 3α2 + 4α3 + 2α4)

7 sinh ε(2α1 + 4α2 + 3α3 + 2α4 + 2α5 + 2α6)

α4 = cosh εα4 + ı/√

7 sinh ε(2α1 + 2α2 + 2α3 + 3α4 + 4α5 + 2α6)

7 sinh ε(2α3 + 4α4 + 3α5 + 2α6),

7 sinh ε(2α1 − 2α3 − 2α4 − 2α5 − α6)

⇒ deformed simple B3-roots as (αij := αi − αj )

β1 = α1 + α6 = cosh ε(α1 + α6)−ı/√

7 sinh ε[3(α1 − α6) + 2(α2 − α5)]

β2 = α2 + α5 = cosh ε(α2 + α5)+ı/√

7 sinh ε[2(α16 + α34) + α25]

β3 = α3 + α4 = cosh ε(α1 + α6)−ı/√

7 sinh ε[2(α2 − α5) + α3 − α4]

CT-symmetrically deformed longest element

• longest element:

w0 : ∆± → ∆∓, w20 = I, αi 7→ −αı = (w0α)i

• representation in terms of Coxeter transformations

σh/2 for h even,σ+σ

(h−1)/2 for h odd.

• action

A` : αı = α`+1−i ,

αı = αi for 1 ≤ i ≤ `, when ` evenαı = αi for 1 ≤ i ≤ `− 2, α¯ = α`−1, when ` odd,

E6 : α1 = α6, α3 = α5, α2 = α2, α4 = α4,B`,C` : αı = αi

E7,E8,F4 : αı = αiG2 : αı = αi

A` :α`α`−1α3α2α1

· · ·u u u u u −w0−→α1α2α3α`−1α`

· · · uuuu uD2`+1 :

α2`+1

α2`−1α3α2α1· · ·

u u u u u

u−w0−→

α2`+1

α2`−1α3α2α1· · ·

u u u u u

α6α5α4α3

uu u u uu −w0−→

uu u u uu α1α3α4α5

• two cases:

[σ, θε] = 0⇒

no solution for h evenprevious solutions for h odd

[σ, θε] 6= 0, θ∗εw0 = w0θε, θ∗ε = θ−1ε , det θε = ±1 lim

ε→0θε = I

Solutions:

∆(ε) for A3

θε =

cosh ε 0 ı sinh ε(− sinh2 ε

2 + ı2 sinh ε) 1 (− sinh2 ε

2 −ı2 sinh ε)

−ı sinh ε 0 cosh ε

∆(ε) for E6

θε =

1 0 0 00 1 0 00 0 θA3

ε 00 0 0 1

PT-symmetrically deformed Weyl reflections

Deformed Weyl reflections

This construction only works for groups of rank 2.

Construction of new models

For any model based on roots, these deformed roots can beused to define new invariant models simply by

α→ α.

For instance Calogero models:

Generalization of Calogero’s solution, undeformed case

• generalized Calogero Hamiltionian (undeformed)

HC(p,q) =p2

∑α∈∆+

(α · q)2 +∑α∈∆+

gα(α · q)2 ,

• define the variables

z :=∏α∈∆+

(α · q) and r2 :=1

∑α∈∆+

(α · q)2,

h ≡ dual Coxeter number, t` ≡ `-th symmetrizer of I• Ansatz:

ψ(q)→ ψ(z, r) = zκ+1/2ϕ(r)

⇒ solution for κ = 1/2√

1 + 4g.

ϕn(r) = cn exp

−√

ht`2ω

√ ht`2ωr2

Lan(x) ≡ Laguerre polynomial, a =

(2 + h + h

√1 + 4g

)l/4− 1

HC(p,q) =p2

∑α∈∆+

(α · q)2 +∑α∈∆+

gα(α · q)2 ,

z :=∏α∈∆+

(α · q) and r2 :=1

∑α∈∆+

(α · q)2,

ψ(q)→ ψ(z, r) = zκ+1/2ϕ(r)

1 + 4g.

ϕn(r) = cn exp

−√

ht`2ω

√ ht`2ωr2

(2 + h + h

√1 + 4g

)l/4− 1

HC(p,q) =p2

∑α∈∆+

(α · q)2 +∑α∈∆+

gα(α · q)2 ,

z :=∏α∈∆+

(α · q) and r2 :=1

∑α∈∆+

(α · q)2,

ψ(q)→ ψ(z, r) = zκ+1/2ϕ(r)

1 + 4g.

ϕn(r) = cn exp

−√

ht`2ω

√ ht`2ωr2

(2 + h + h

√1 + 4g

)l/4− 1

• eigenenergies

En =14

[(2 + h + h

√1 + 4g

)l + 8n

]√ ht`2ω

• anyonic exchange factors

ψ(q1, . . . ,qi ,qj , . . .qn) = eıπsψ(q1, . . . ,qj ,qi , . . .qn), for 1 ≤ i , j ≤ n,

withs =

√1 + 4g

∵ r is symmetric and z antisymmetric

The construction is based on the identities:∑α,β∈∆+

α · β(α · q)(β · q)

=∑α∈∆+

(α · q)2 ,

∑α,β∈∆+

(α · β)(α · q)

(β · q)=

t`,∑α,β∈∆+

(α · β) (α · q)(β · q) = ht`∑α∈∆+

(α · q)2,

∑α∈∆+

α2 = `ht`.

Strong evidence on a case-by-case level, but no rigorous proof.

Antilinearly deformed Calogero Hamiltionian

• antilinearly deformed Calogero Hamiltionian

HadC(p,q) =p2

∑α∈∆+

(α · q)2 +∑α∈∆+

gα(α · q)2

z :=∏α∈∆+

(α · q) and r2 :=1

∑α∈∆+

(α · q)2

• Ansatzψ(q)→ ψ(z, r) = zsϕ(r)

when identies still hold⇒

ψ(q) = ψ(z, r) = zsϕn(r)

with same eigenenergies

HadC(p,q) =p2

∑α∈∆+

(α · q)2 +∑α∈∆+

gα(α · q)2

z :=∏α∈∆+

(α · q) and r2 :=1

∑α∈∆+

(α · q)2

HadC(p,q) =p2

∑α∈∆+

(α · q)2 +∑α∈∆+

gα(α · q)2

z :=∏α∈∆+

(α · q) and r2 :=1

∑α∈∆+

(α · q)2

HadC(p,q) =p2

∑α∈∆+

(α · q)2 +∑α∈∆+

gα(α · q)2

z :=∏α∈∆+

(α · q) and r2 :=1

∑α∈∆+

(α · q)2

Deformed A3-models

• potential from deformed Coxeter group factorsα1 = 1,−1,0,0, α2 = 0,1,−1,0, α3 = 0,0,1,−1

α1 · q = q43 + cosh ε(q12 + q34)− ı√

2 cosh ε sinhε

2(q13 + q24)

α2 · q = q23(2 cosh ε− 1) + ı2√

2 cosh ε sinhε

α3 · q = q21 + cosh ε(q12 + q34)− ı√

2 cosh ε sinhε

2(q13 + q24)

α4 · q = q42 + cosh ε(q13 + q24) + ı√

2 cosh ε sinhε

2(q12 + q34)

α5 · q = q31 + cosh ε(q13 + q24) + ı√

2 cosh ε sinhε

2(q12 + q34)

α6 · q = q14(2 cosh ε− 1)− ı√

2 cosh ε sinhε

notation qij = qi − qj , No longer singular for qij = 0

Deformed A3-models

• potential from deformed Coxeter group factorsα1 = 1,−1,0,0, α2 = 0,1,−1,0, α3 = 0,0,1,−1

α1 · q = q43 + cosh ε(q12 + q34)− ı√

2 cosh ε sinhε

2(q13 + q24)

α2 · q = q23(2 cosh ε− 1) + ı2√

2 cosh ε sinhε

α3 · q = q21 + cosh ε(q12 + q34)− ı√

2 cosh ε sinhε

2(q13 + q24)

α4 · q = q42 + cosh ε(q13 + q24) + ı√

2 cosh ε sinhε

2(q12 + q34)

α5 · q = q31 + cosh ε(q13 + q24) + ı√

2 cosh ε sinhε

2(q12 + q34)

α6 · q = q14(2 cosh ε− 1)− ı√

2 cosh ε sinhε

notation qij = qi − qj , No longer singular for qij = 0

• PT -symmetry for α

σε− : α1 → −α1, α2 → α6, α3 → −α3, α4 → α5, α5 → α4, α6 → α2

σε+ : α1 → α4, α2 → −α2, α3 → α5, α4 → α1, α5 → α3, α6 → α6

• PT -symmetry in dual space

σε− : q1 → q2, q2 → q1, q3 → q4, q4 → q3, ı→ −ıσε+ : q1 → q1, q2 → q3, q3 → q2, q4 → q4, ı→ −ı

σε−z(q1,q2,q3,q4) = z∗(q2,q1,q4,q3) = z(q1,q2,q3,q4)

σε+z(q1,q2,q3,q4) = z∗(q1,q3,q2,q4) = −z(q1,q2,q3,q4)

ψ(q1,q2,q3,q4) = eıπsψ(q2,q4,q1,q3).

σε− : α1 → −α1, α2 → α6, α3 → −α3, α4 → α5, α5 → α4, α6 → α2

σε+ : α1 → α4, α2 → −α2, α3 → α5, α4 → α1, α5 → α3, α6 → α6

ψ(q1,q2,q3,q4) = eıπsψ(q2,q4,q1,q3).

σε− : α1 → −α1, α2 → α6, α3 → −α3, α4 → α5, α5 → α4, α6 → α2

σε+ : α1 → α4, α2 → −α2, α3 → α5, α4 → α1, α5 → α3, α6 → α6

ψ(q1,q2,q3,q4) = eıπsψ(q2,q4,q1,q3).

Anyonic exchange factors in the 4-particle scattering process

u u u uw x y z

q1 q2 q3 q4= eıπs u u u uw x y z

q2 q4 q1 q3

u uu ux y z

q2 = q3q1 q4= eıπs u uu ux y z

q2 q1 = q4 q3

uu uux y

q1 = q2 q3 = q4= eıπs uu uux y

q1 = q3 q2 = q4

uuu ux y

q1 = q2 = q3 q4= u uuux y

q4 q1 = q2 = q3

u u u uw x y z

q2 q4 q1 q3

u uu ux y z

q2 q1 = q4 q3

uu uux y

q1 = q3 q2 = q4

uuu ux y

q4 q1 = q2 = q3

u u u uw x y z

q2 q4 q1 q3

u uu ux y z

q2 q1 = q4 q3

uu uux y

q1 = q3 q2 = q4

uuu ux y

q4 q1 = q2 = q3

u u u uw x y z

q2 q4 q1 q3

u uu ux y z

q2 q1 = q4 q3

uu uux y

q1 = q3 q2 = q4

uuu ux y

q4 q1 = q2 = q3

• Physical properties of the A2,G2-models:The deformed model can be solved by separation ofvariables as the undeformed case.Some restrictions cease to exist, as the wavefunctions arenow regularized.⇒ modified energy spectrum:

E = 2 |ω| (2n + λ+ 1)

becomes

E±n` = 2|ω|[2n + 6(κ±s + κ±l + `) + 1

]for n, ` ∈ N0,

with κ±s/l = (1±√

1 + 4gs/l)/4

Conclusions

ConclusionsWe provided antilinear deformations of Coxeter groupsInvariant Coxeter models can be constructed with newphysical propertiesOpen issues:

mathematical constructionmost general solution, closed formulae for infinite seriesrelaxing some constraintssolutions for unsolvable casesother elements inW

Calogero modelsproof for identitiesfurther solutions to Schrödinger equationHermitian counterparts

integrability of deformed models?application to other models

Thank you for your attention.

Conclusions

Antilinear deformations of Coxeter groups, an …fring/Micro.pdf · Introduction Invariant Calogero...

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